So theoretically, the NextBox could run it much better if it has BC? Assuming it doesn't use emulation?

Possibly, but I wouldn't count on it. A huge part of the problem is how compressed the data is; remember that a 360 disc can technically only hold 4.3 gigs of data (same a DVD), and the PC build of Skyrim is about 10 gigs. So even with more console RAM in place there are still likely to be freezes due to the console being unable to properly decode the correct processes within the specified time frame. Most lock-up freezes happen when something within the cell has changed (i.e. an enemy is spawning), or you are moving between cells. A required code string isn't found right away, other code strings continue to load and activate, and then it becomes too late - even when the correct process is found, everything else has skipped ahead too far and now it's out of sync. You can get similar crashes on a PC if you're using a lot of mods which have added all sorts of overlapping code.

Having more RAM in a next gen console will help with retrieval speed and should reduce lock up crashes, but you'll never completely eliminate the time required to decompress the files, so some freezing will still take place.

I started to have freezing issues on load screens after I downloaded the DLC's. I have now been in the practice of clearing the cache out when I load the game. This seems to have cured my freezing problem.

To clear the cache RB + LB + "X" from as soon as I see the Besthada logo until I get the main menu.